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In this article we exhibit certain projective degenerations of smooth $K3$surfaces of degree $2g-2$ in $\Bbb P^g$ (whose Picard group is generated by thehyperplane class), to a union of two rational normal scrolls, and also to aunion of planes. As a consequence we prove that the general hyperplane sectionof such $K3$ surfaces has a corank one Gaussian map, if $g=11$ or $g\geq 13$.We also prove that the general such hyperplane section lies on a unique $K3$surface, up to projectivities. Finally we present a new approach to theclassification of prime Fano threefolds of index one, which does not rely onthe existence of a line.Comment: 24 pages, AMS-LaTeX 1.

Projective degenerations of K3 surfaces, Gaussian maps, and Fano threefolds